{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0016",
  "pilot_number": "R0045-23491",
  "number": 23491,
  "family_id": "qf-mod43-cauchy-prime-divisor-qualification",
  "title": {
    "zh-Hant": "第 23491 題：有限群的Cauchy定理",
    "en": "Question 23491: Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0016",
    "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-cauchy-theorem-finite-groups"
    ],
    "title_zh": "有限群的Cauchy定理",
    "stem_zh": "完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)",
      "explanation_zh": "Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限群的Cauchy定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod43-cauchy-prime-divisor-qualification",
      "seed": 202611121016,
      "parameters": {
        "group_table": [
          [
            3,
            4,
            5,
            0,
            1,
            2
          ],
          [
            4,
            5,
            0,
            1,
            2,
            3
          ],
          [
            5,
            0,
            1,
            2,
            3,
            4
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            2,
            3,
            4,
            5,
            0
          ],
          [
            2,
            3,
            4,
            5,
            0,
            1
          ]
        ],
        "identity": 3,
        "prime": 2,
        "domain": "Complete_finite_group_Cauchy_witness",
        "operation": "cauchy-prime-divisor-qualification",
        "semantic_template": "qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"prime\":2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0016",
    "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e588b78591b9fbe5e9e4c08a523f93cf138d3732a59377822e230796a42c550c",
      "family_id": "qf-mod43-cauchy-prime-divisor-qualification",
      "seed": 202611121016,
      "semantic_template": "qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"prime\":2}"
    },
    "fields": {
      "title": "Cauchy Requires a Prime Divisor of a Fully Certified Finite Group",
      "stem": "Use complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)",
        "explanation": "Require primality and prime divisibility of the complete group order. Qualification is separate from producing an exact-order element witness. Finite model enumeration is not a proof of the general Cauchy theorem. The exact result, retaining the requested order, is (6, 2, Yes, 0, complete_group_and_prime_divisibility_hypotheses_hold)."
      },
      "skills": [
        "Cauchy Requires a Prime Divisor of a Fully Certified Finite Group"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]] with identity=3, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Declared prime=2. Give (group_order, prime, qualified, divisibility_remainder, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=cauchy-prime-divisor-qualification;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[group_order, prime, qualified, divisibility_remainder, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。",
        "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL closed Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]];ACTUAL declared group identity=3",
      "after": "ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]];two-sided group identity=是",
      "relation": "implies",
      "rule_id": "groupaction_group_identity_both_sides_not_action_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "群表的e*g與g*e兩側都須等g；這裡不把作用點的e*x=x當群identity證書。",
        "en": "Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, 是], [1, 0, 0, 是], [2, 1, 1, 是], [3, 2, 2, 是], [4, 3, 3, 是], [5, 4, 4, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 2, 2, 是], [2, 3, 3, 是], [3, 4, 4, 是], [4, 5, 5, 是], [5, 0, 0, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, 是], [1, 3, 3, 是], [2, 4, 4, 是], [3, 5, 5, 是], [4, 0, 0, 是], [5, 1, 1, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, 是], [1, 4, 4, 是], [2, 5, 5, 是], [3, 0, 0, 是], [4, 1, 1, 是], [5, 2, 2, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(6)",
      "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, 是], [1, 5, 5, 是], [2, 0, 0, 是], [3, 1, 1, 是], [4, 2, 2, 是], [5, 3, 3, 是]]",
      "relation": "implies",
      "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。",
        "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ORIGINAL group element g=0;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, 是], [1, 4, 4, 否], [2, 5, 5, 否], [3, 0, 0, 否], [4, 1, 1, 否], [5, 2, 2, 否]]",
      "after": "first actual two-sided inverse witness=0;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ORIGINAL group element g=1;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, 否], [1, 5, 5, 否], [2, 0, 0, 否], [3, 1, 1, 否], [4, 2, 2, 否], [5, 3, 3, 是]]",
      "after": "first actual two-sided inverse witness=5;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ORIGINAL group element g=2;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, 否], [1, 0, 0, 否], [2, 1, 1, 否], [3, 2, 2, 否], [4, 3, 3, 是], [5, 4, 4, 否]]",
      "after": "first actual two-sided inverse witness=4;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ORIGINAL group element g=3;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, 否], [1, 1, 1, 否], [2, 2, 2, 否], [3, 3, 3, 是], [4, 4, 4, 否], [5, 5, 5, 否]]",
      "after": "first actual two-sided inverse witness=3;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "ORIGINAL group element g=4;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, 否], [1, 2, 2, 否], [2, 3, 3, 是], [3, 4, 4, 否], [4, 5, 5, 否], [5, 0, 0, 否]]",
      "after": "first actual two-sided inverse witness=2;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL group element g=5;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, 否], [1, 3, 3, 是], [2, 4, 4, 否], [3, 5, 5, 否], [4, 0, 0, 否], [5, 1, 1, 否]]",
      "after": "first actual two-sided inverse witness=1;exists=是",
      "relation": "implies",
      "rule_id": "groupaction_each_element_actual_two_sided_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。",
        "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "structurally closed COMPLETE group candidate;two-sided identity=是;ALL-triple associativity=是;ALL-element inverses=是",
      "after": "group_valid=是 is the CONJUNCTION;False remains a reported candidate result,not a silent repair",
      "relation": "implies",
      "rule_id": "groupaction_group_axiom_conjunction_candidate_may_fail",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整三項責任合取才是群資格。即使作用表看似排列，壞group table也不會因此變群；不自行換identity或修表。",
        "en": "Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "CURRENT ORIGINAL complete Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]];actual identity=3;order=6",
      "after": "ALL group axioms qualified;actual inverse list=[0, 5, 4, 3, 2, 1];no point carrier or group-action assumption is used",
      "relation": "implies",
      "rule_id": "group_CURRENT_complete_certified_Cayley_NOT_action_wrapper",
      "conditions": [],
      "explanation": {
        "zh-Hant": "群表本身先完整驗閉合、identity、associativity、雙邊inverse；不套舊作用表的8元素限制，也不假設cyclic或commutative。純舊函式只代入本題表與trace。",
        "en": "Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Cauchy ORIGINAL prime divisor:ORIGINAL integer=2;complete possible trial divisors=[]",
      "after": "ALL [divisor,remainder] rows=[];prime=是;a composite>1 has a factor at most its square root",
      "relation": "implies",
      "rule_id": "finite_prime_ALL_trial_divisors_through_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整平方根範圍核整除；prime2的範圍為空而通過，0/1不通過。不能由少數取樣、平方表或Euler判準代替此資格。",
        "en": "Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "CURRENT qualified group order=6;ORIGINAL prime p=2;divisibility remainder=0",
      "after": "product-e p-tuples have count |G|^(p-1)=6^1;cyclic rotation preserves product e by conjugation;rotation orbits have size1 or p;fixed tuples are(g,...,g) with g^p=e",
      "relation": "implies",
      "rule_id": "cauchy_prime_divisibility_and_general_rotation_count_not_enumeration",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前p−1個元素可任選，最後一個由inverse唯一決定。rotation後乘積是首元素逆乘原乘積再乘首元素，仍為e，不偷換非交換次序。prime使周期只能1或p；總數被p整除，所以fixed解數也被p整除，含identity便至少p個，故有非identity解；其精確階整除prime而只能p。這是一般論證，不實際枚舉天文數量tuple。",
        "en": "Choose the first p−1 elements freely;the last is uniquely the inverse product.Rotation conjugates the original product,so preserves e without commuting factors.Primality makes periods1 or p;the total count is divisible by p,so the fixed-solution count is too.Including the identity gives at least p solutions and a nonidentity one whose exact order divides p and hence equals p.This is a general argument,not enumeration of astronomical tuples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      6,
      2,
      true,
      0,
      "complete_group_and_prime_divisibility_hypotheses_hold"
    ]
  },
  "conclusion": {
    "canonical_value": "(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations/questions.jsonl",
      "batch_sha256": "fab0187f15949aba06bf6ee6e60f81983e81907a1fb62931b50bc84c8521e4c6",
      "original_line": 16,
      "original_line_sha256": "538b87229490dabd1b8d85ad4a72795c41bca81d9df1e62e3748d754d0538610",
      "question_compact_sha256": "e588b78591b9fbe5e9e4c08a523f93cf138d3732a59377822e230796a42c550c",
      "english_row_compact_sha256": "abec6022f361e51c7676a0ad80bf1b69e1e8ff7cf53588ad7ab0a8fdc38edc90",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-cauchy-theorem-finite-groups": {
        "sha256": "779e8ef831907c123fc1e7039941b328b38b54af646d6a171eab7e28b476927d",
        "english_sha256": "557db8e818ba9af78d5bbcd15fb786238f5f2099b296f105fa26a68e62fd821f",
        "original_path": "public/data/mko/cauchy-theorem-finite-groups.json",
        "english_path": "public/data/i18n/editions/mko-cauchy-theorem-finite-groups.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8",
    "original_candidate_sha256": "f54197847e653b0f10142001322ac9d7c54bb3b25962da2000d949061702932a",
    "original_candidate_line_sha256": "f54197847e653b0f10142001322ac9d7c54bb3b25962da2000d949061702932a"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p46",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438",
      "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 48,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8b9004877c7e04123374c55c792db7d43cfeb0f0bd3d7c386c7efee9cd8f62d9",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}